I’ve been reading a lot of thought-provoking stuff lately but haven’t processed it enough to write about it yet. A lot of it is about “Mathematical Knowledge for Teaching” which is kind of a hip idea in math education research – the specifically mathematical, as opposed to pedagogical, knowledge that we need to do our job, that is different from what other professionals in other mathematically-intense professions (academic mathematical research, risk management, physics/engineering/chemistry, statistics, etc.) need. I look forward to sharing some of this soon.

In the meantime – my humble attempt at a post in the format that make f(t) and Continuous Everywhere so great – some reflection, mid-stream, on a new structure I’m trying out. The circumstances are very unusual but maybe the ideas will still be useful.

So, currently, the only actual class I’m teaching is an informal seminar on group theory for some friends (most are high school math teachers). We’re meeting weekly; our plan is to go thru the rest of the school year. There are about 4-5 folks who are coming consistently, with a few more intermittent folks.

In the context of this informality, we sort of spontaneously began a pedagogical/technological experiment that I’m excited about and hope to refine and use again in more formal settings: we’re maintaining a group-authored Google Doc that contains summaries of all the content to date, and we’ve also just begun a Google Wave to facilitate discussion about the course content between meetings.

My excitement notwithstanding, our implementation has a lot of chinks to work out.

I proposed the document with summaries on the first day of class. My idea was a) to create a record that would allow folks who missed a session to see what they missed; b) promote ownership of the content by having particpants take turns producing the summaries; and c) at the end, we’d have this awesome record of the course, to solidify what we’d accomplished. It was a participant’s idea to do it as a google doc rather than just having people email me the summaries.

In the beginning, I’d take a volunteer at the end of each session to write it up for the following session. Plus: people tended to write focused, coherent summaries that seemed to me to be a pretty good record for somebody who missed class of what we’d done. Minus: it was hard to use them for this because people tended not to get the summary up until the day before the next class, so if you missed class you pretty much went to the next class not having read what happened. Plus: I think these summaries will be nice to have at the very end. Minus: the goal of group ownership wasn’t getting served by having just one person write something each week. People didn’t really seem to feel like writing up the notes was an opportunity to gain understanding of the content; just a task to get done. (And these are folks who spend their professional lives thinking about what people have to do to gain understanding of content.)

On I think the fifth session (we’ve had 7 total), one of the participants suggested that we all add to the summary, to give us a chance to take ownership of it as a class to a greater extent.

It did feel like this led to more engagement with the Doc. Three or four people contributed to the summary that week. On the other hand, the summary immediately became much more internal – people were talking with each other, referring to things that had happened in class – you couldn’t really follow it if you hadn’t been there in class, and I feel pretty sure that week’s summary will be hard to understand by May. It feels a lot less complete and focused and has a strong “you had to be there” vibe.

In response to this, we decided to start a Google Wave to house the conversation between folks, and continue to maintain the Doc as a repository for a polished summary. For those who (like me, a week and a half ago) don’t know, Google Wave is a glorified chat program. The conversations are saved and can be edited. You can start new ones, edit other people’s comments, etc. I don’t really know much about it yet but one of the participants was really excited about it.

In the first week and a half of the Wave’s existence, what has happened is that one of the participants (who is a mathematics PhD and knows more about almost everything, except for the one subject I happen to be teaching, than I do) got really interested in a question I posed, and he and I got into a long conversation about it, which nobody else participated in, and nobody else said anything about anything else either.

My thoughts about the experience so far:

I mean in principal it’s awesome, right? Participants are producing their own coauthored textbook on group theory, and have the opportunity to bang out the details of their understanding in live dialogue with each other during the week between classes! … uh,
… uh … not if nobody’s on there foo.

One thought is, if in the future I’m teaching a course like this for credit, then it could just be a course requirement to do the summary for the Doc at least once, and to make a comment or ask a question on the Wave at least once each week. I fantasize that forcing people to get on the Wave would make it blossom into a full-blown orgy of mathematical debate.

Granted, the circumstances under which I’m doing this make it hard for me to believe that it’s even consistently happening, let alone that people would be putting in the additional time to really make our little Doc/Wave experiment come alive. Every time four or five busy adults, classroom teachers and mathematicians at that, trek out on a school night to do group theory with me, for no kind of credit or any certified acknowledgement of any kind, I feel grateful. That said, if these folks are dedicated enough to understanding group theory to show up, and it will definitely improve their understanding to talk with each other about it online between classes, it seems worth it to try to figure out how to make the experiment come alive.

Ideas going forward:

I’ve been shy to edit folks’ contributions to both the Doc and the Wave, although once or twice I deleted or glossed vocabulary used by some participant that I didn’t think everyone would know. But I think I need to be more forceful in making sure everyone’s contributions are accessible to everyone else (or are happening out of the others’ view). It needs to be totally easy to slip into the conversation. The crowd is an extremely mathematically empowered crowd overall, but the people most active on the Wave (and in the last 2 weeks on the Doc) are the really, extra extremely mathematically empowered ones; and they’ve been dropping some not-necessarily-generally-familiar vocabulary. I have to rein this in.

Actually probably the most straightforwardly productive thing to do is just to ask the group what they want from the Wave and how we could achieve that. If it’s not to be, it’s not to be, but if it is going to be awesome, it is going to be so precisely because the group takes ownership of it and it’s not just me cheerleading.

Recently I had a conversation with a teacher I work with, who teaches 10th grade geometry, that led me to a clearer articulation of something I started to try to say before. Namely:

The teaching of proof needs to be connected to kids’ own sense of what they are sure of.

This is actually obvious when you think about it. How could proof – the art and science of coming to know things for sure – be learned if the distinction between what the learner does and does not know for sure is not involved in the process?

I’ve just started reading an article that appears to be suggesting that this claim is also supported by research (more below). And yet this is not typically how proof is taught.

First of all, a typical proof problem in a geometry textbook is asking you to prove something that, as I’ve discussed before, is just about as visually clear as the givens are. So there’s really nothing you’re not sure of at all; and the process must proceed totally disconnected to your own sense of what you actually feel confident about. (Hence the kids come up with these arguments that follow the two-column format but don’t make any sense. I speak from experience. If you’ve taught a “proof unit” in geometry or algebra, you know what I’m talking about.)

But more broadly, in geometry and other classes: any problem of the form “prove X” is a setup for kids to fail to understand proof. It’s a fine kind of problem for someone who already really understands what proof is all about, so go ahead with this kind of problem in graduate classes or anywhere else if you feel your students have a well-developed sense of rigor. (And telling them what to prove is a great hint to make the problem easier without giving away too much.) BUT:

If a student does not yet understand proof, in the sense that she cannot yet produce coherent proofs, problems of the form “prove X” are not what to learn on. Why? Because once you say “prove X”, this student already believes X. You said she was supposed to prove it so it must be true.

This robs the student of her ability to sense and be guided by what she is or isn’t sure of.

Without this sense as a guide, proof becomes a shell game, and one whose rules are insubstantial and shifting, because they aren’t really the rules. There is really only one rule to proof: it has to be convincing. This is the guide and the judge. If you already “know” X is true (because the teacher told you to prove it), the guide and the judge automatically take a lunch break – you are already convinced. All you can do is write some things down in the “reasons” column and hope your teacher likes them.

What I think a “well-developed sense of rigor” really is, is the habit of bracketing anything you haven’t been totally convinced of as different from anything you have. So to a student who has such a “well-developed sense of rigor” you can say “prove X” and X is still in brackets, so she can still head toward it guided by the goal of removing the brackets. She trusts the authority of her own reasoning.

But this is not the state of most kids I have taught, or seen taught, at the K-12 level. So “prove X” is the wrong problem. “Prove or disprove X” is always better. (Unfortunately it is also always harder, and may therefore be too hard when “prove X” wouldn’t have been. But the comparative easiness isn’t worth it. They need problems where they actually don’t know the truth and have to figure it out for themselves. Otherwise they don’t learn how to prove. We have to find the X so that “prove or disprove X” is at the right level of difficulty.)

The paper I alluded to is from the May 2002 Journal of Research in Mathematics Education. It’s by Patricio Herbst and is called “Engaging Students in Proving: A Double Bind on the Teacher”. I just started it, so can’t really tell you what it’s about with confidence yet. But it contains the sentence “Studies of how students prove have demonstrated the importance, from the perspective of students’ learning, of maintaining the connections between proving and knowing (Balacheff, 1987, 1990, 1991; Chazan, 1993; Senk, 1989).” (p. 177) I take this (partly based on context) to be making my exact point, but I haven’t followed up with the citations yet. Unfortunately several of Balacheff’s are in French, which I don’t read.